Henstock–Kurzweil integral

E259763

generalized Riemann integral integration theory mathematical integral

The Henstock–Kurzweil integral is a highly general integration theory that extends and refines the Riemann integral, capable of integrating a broader class of functions while retaining many of the intuitive properties of Riemann integration.

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All labels observed (3)

Label	Occurrences
Denjoy integral	1
Henstock–Kurzweil integral canonical	1
generalized Riemann–Henstock integral	1

Statements (49)

Predicate	Object
instanceOf	generalized Riemann integral ⓘ integration theory ⓘ mathematical integral ⓘ
admitsExtensionTo	improper integrals over unbounded intervals ⓘ
allows	fine partitions adapted to local behavior of the integrand ⓘ
alsoKnownAs	gauge integral ⓘ Henstock–Kurzweil integral ⓘ surface form: generalized Riemann–Henstock integral
canIntegrate	all Lebesgue integrable functions ⓘ all improper Riemann integrable functions ⓘ some functions not Lebesgue integrable ⓘ
characterizedBy	gauge condition on partitions instead of uniform mesh size ⓘ
compatibleWith	classical techniques of Riemann integration ⓘ
contrastsWith	measure-theoretic approach of Lebesgue integration ⓘ
definitionBasedOn	gauges ⓘ tagged partitions ⓘ
domain	real-valued functions on intervals of the real line ⓘ
extends	Lebesgue integration ⓘ surface form: Lebesgue integral Riemann integral ⓘ
field	real analysis ⓘ
generalizes	Lebesgue integral on measurable functions ⓘ Riemann integral on bounded intervals ⓘ
hasProperty	every Lebesgue integrable function has the same Henstock–Kurzweil and Lebesgue integrals ⓘ every derivative is Henstock–Kurzweil integrable ⓘ integral is unique when it exists ⓘ integral of a derivative recovers the original function up to a constant under mild conditions ⓘ
hasTextbookTreatmentIn	Jaroslav Kurzweil’s works on generalized integration ⓘ Ralph Henstock’s books on non-absolute integration ⓘ
implies	Denjoy integral on many functions ⓘ
includesAsSpecialCase	Riemann–Stieltjes integral for suitable integrators ⓘ
is	nonabsolute integral ⓘ
isDefinedOn	bounded intervals of the real line ⓘ
isEquivalentTo	Denjoy integral for functions with certain regularity conditions ⓘ
isNot	countably additive measure integral ⓘ
isStrongerThan	Lebesgue integral in terms of integrable functions ⓘ
isWeakerThan	Denjoy integral in generality ⓘ
namedAfter	Jaroslav Kurzweil ⓘ Ralph Henstock ⓘ
preservesProperty	fundamental theorem of calculus for all differentiable functions with regulated derivative ⓘ
relatedConcept	fine partition ⓘ gauge ⓘ nonabsolute integration ⓘ
retains	intuitive partition-sum interpretation of the integral ⓘ
satisfies	additivity over intervals ⓘ linearity ⓘ translation invariance on the real line ⓘ
subsumes	Lebesgue integration ⓘ surface form: Lebesgue integral on the real line
usedIn	probability theory and stochastic integration generalizations ⓘ theory of differential equations ⓘ
yearIntroduced	1957 ⓘ

Referenced by (3)

Full triples — surface form annotated when it differs from this entity's canonical label.

Riemann integral → contrastedWith → Henstock–Kurzweil integral ⓘ

Arnaud Denjoy → knownFor → Henstock–Kurzweil integral ⓘ

this entity surface form: Denjoy integral

Henstock–Kurzweil integral → alsoKnownAs → Henstock–Kurzweil integral ⓘ

this entity surface form: generalized Riemann–Henstock integral